Only answer part E.
3. The Cheshire Kid has been up to no good since we last had our eye on him. He's planning a daring heist, and he's assembled himself a team of notorious outlaws: Pinball Pete 6, wanted in three states for hustling unsuspecting pool hall patrons, and Jackrabbit Jill 7, who's hopped out of custody of more federal marshals than the number of needles on a prickly pear! Together, the three of them intend to rob a train that is due to leave the town of Sleepy Eye any second now. The three have set up camp a mile outside of town, with a clear view of the train station. They take out their binoculars and observe the train as it leaves. Their plan is to wait for the train to get far enough away from town so they can intercept it and jump onto the train without being seen. 8 Once they're in, they plan to help themselves to all the gold bars and bearer bonds 9 they can carry. Let D(t) be the distance between the three bandits and the train t minutes after the train leaves the station. In this case, we have that D(t)=2e0.1t1 The Cheshire Kid and his accomplices wish to intercept the train 10 minutes after it leaves the station, so they'll need to estimate how far the train will have traveled in that time so they can head it off. Unfortunately, none of the three bandits has a calculator with them to determine D(10) exactly. Fortunately, seeing as they do know some differential calculus, they know that they can estimate D(t) using simpler functions. (a) Unlike our three unscrupulous protagonists, we do have access to calculators. Draw the following table and fill in the missing values: (b) Since D(0) exists, we can use the line tangent to D(t) at t=0 (which we'll call L(t) ) to approximate values for D(t) near zero. Find an equation for L(t). Add a third row to your table from part (a) and fill in the corresponding values for L(t). (c) Let E1(t)=D(t)L(t) represent the magnitude of the error of the linear approximation L(t). Calculate E1(t) for each column in the table. (d) Not only does D(0) exist in this case, but so does D(0). For this reason, we can improve upon our linear approximation L(t) using the quadratic approximation Q(t) of D(t) : Q(t)=L(t)+2D(0)t2 Add a fourth row to your table from part (a), fill it with the corresponding values of Q(t), and for each of these values find the magnitude of the error, E2(t)=D(t)Q(t). (e) How do the errors of the two approximations compare? Could we potentially improve the accuracy of our approximation by adding another term? What might that term look like? What kind of information would we need to know about D(t) in order to find it
